A typical vehicle automatic transmission includes at least one input clutch unit contained within a generally cylindrical housing through which a central shaft of the transmission coaxially extends. The input clutch unit includes a clutch plate pack made up of a first set of plates splined at their outer diameter to the inside of the housing and an interleaved second set of plates splined at their inner diameter to the outside of the outer race of an overrunning clutch. The overrunning clutch is located in the annular space between the clutch plate pack and the central shaft. A piston alternately forces the two sets of clutch plates together or releases them to selectively vary the power flow path through the transmission. The overrunning clutch serves as a speed sensor that matches the relative rotation of the clutch pack to the rotation of the central shaft so as to give a smoother transfer of power. When the two sets of clutch plates are released, they rotate very rapidly and closely relative to one another, which requires that a cooling lubricant be supplied between them to avoid overheating. That lubricant is typically supplied to the clutch plate pack, as well as to other components of the transmission, by being pumped down passages extending through or surrounding the central shaft. The lubricant is then directed radially outwardly to the clutch plate pack, either by pumping pressure, or by the centrifugal force of the overrunning clutch outer race, which rotates very rapidly whenever such lubricant is required, i.e., when the clutch plate pack is released. The lubricant is typically reaches the plate pack through holes drilled through the outer race, which holes provide a short, direct path. However, in order to deliver enough lubricant, a great number of such lubricant supply holes are often needed, and these must each be separately drilled in the outer race, which is expensive and time consuming. U.S. Pat. No. 4,714,803, assigned to the assignee of the current invention, discloses a roller clutch cage and seal combination that better confines lubricant between the clutch races, so as to more efficiently deliver lubricant through the outer race supply holes. While this can potentially reduce the number or size of such supply holes, it does not eliminate them altogether.